# Two-Stage Robust Optimal Scheduling of Flexible Distribution Networks Based on Pairwise Convex Hull

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## Abstract

**:**

## 1. Introduction

## 2. Deterministic Optimal Scheduling for FDNs

#### 2.1. Objective Function

_{i}

_{,t}is the real power injection of bus i during time period t; ${P}_{i,t,\mathrm{FDS}}^{\mathrm{loss}}$ is the VSC power loss of the multiterminal FDS near the terminal of bus i during time period t; ${P}_{i,t,\mathrm{ESS}}^{\mathrm{loss}}$ is the charging and discharging power loss of the ESS installed at bus i during time period t; Ω

_{T}is the set of all time periods; Ω

_{b}is the set of all buses; Ω

_{FDS}is the set of all FDSs; Ω

_{b(v)}is the set of all buses associated with the vth FDS; and Ω

_{ESS}is the set of all buses with ESS installed; Δt is the interval (h) of each time period.

#### 2.2. Constraints

#### 2.2.1. Steady-State Operational Constraints of FDSs

_{i}

_{,t,FDS}and Q

_{i}

_{,t,FDS}are the real and reactive power injections of the FDS into bus i during time period t, respectively; A

_{loss,FDS}is the FDS loss factor; Q

_{min,FDS}and Q

_{max,FDS}are the upper and lower limits of the reactive power through each FDS, respectively; and S

_{max,FDS}is the maximum apparent power allowed through the FDS. Equation (3) makes the sum of the real power injection into all associated feeders by the FDS and the power loss of the FDS come out to zero. Equation (4) makes the reactive power injection of the FDS not exceed its adjustable reactive power limit. Equation (5) makes the apparent power of the FDS not exceed its capacity.

#### 2.2.2. Steady-State Operational Constraints of ESSs

_{i,t}is the remaining energy of the ESS installed at bus i during time period t; ${\eta}_{i}^{c}$ and ${\eta}_{i}^{d}$ are the charging and discharging efficiency of the ESS installed at bus i, respectively; E

_{i}

_{,ESS}is the energy capacity of the ESS installed at bus i; and SOC

_{i}

_{,max}and SOC

_{i}

_{,min}are the maximum and minimum state of charge (SOC) of the ESS installed at bus i. Equation (6) makes the charging and discharging power not exceed the maximum value at any time period; Equation (7) makes the remaining energy satisfy the continuity constraint. Equation (8) makes the remaining energy at the end of each day equal to the initial energy of that day. Equation (9) makes the ESS free from deep charging or discharging.

#### 2.2.3. Operational Constraints of OLTCs

_{ij}is the ratio of the OLTC branch i-j; τ

_{ij}

_{,min}is the minimum ratio of the OLTC branch i-j; N

_{ij}

_{,t}is the tap position of the OLTC branch i-j during time period t; N

_{ij}

_{,max}is the maximum tap position number of the OLTC branch i-j; β

_{ij}

_{,max}is the maximum allowed number of daily actions of the OLTC branch i-j; and Δτ

_{ij}is the difference in the ratios between adjacent tap positions of the OLTC branch i-j. For example, if there are five upper tap positions and five lower tap positions, then N

_{ij}

_{,max}= 11, N

_{ij}

_{,t}takes the integers 0~10 and Δτ

_{ij}takes the values 0.01~0.10.

#### 2.2.4. Power Flow Constraints

_{ij}

_{,t}, Q

_{ij}

_{,t}and I

_{ij}

_{,t}are the real and reactive power at the “from” end and current amplitude of branch i-j during time period t, respectively; P

_{i}

_{,t,DG}and P

_{i}

_{,t,L}are the DG real power injection and real power load at bus i during time period t, respectively; Q

_{i}

_{,t,DG}and Q

_{i}

_{,t,L}are the DG reactive power injection and reactive power load at bus i during time period t, respectively; r

_{ij}and x

_{ij}are the resistance and reactance of branch i-j, respectively; and Ω

_{l}is the set of all lines.

#### 2.2.5. Thermal Limit Constraint

_{ij}

_{,max}is the ampacity of branch i-j.

#### 2.2.6. Bus Voltage Constraints

#### 2.3. Model Reformulation as a MISOCP Problem

#### 2.3.1. Reformulation of the Power Flow Constraints

_{ij}

_{,t}and u

_{i}

_{,t}are introduced to replace ${I}_{ij,t}^{2}$ and ${V}_{i,t}^{2}$, and with the help of the big-M relaxation technique, the last two equations in Equation (13) become:

#### 2.3.2. Reformulation of the FDS Constraints

#### 2.3.3. Reformulation of the OLTC Constraints

_{ij}

_{,s,t}is introduced as the flag of the sth tap position for the OLTC branch i-j during time period t. Then, N

_{ij}

_{,t}can be expressed as the cumulative sum of the B

_{ij}

_{,s,t}for each tap position, and the OLTC constraints become:

_{ij}

_{,s,t}is the flag of the sth tap position for the OLTC branch i-j during time period t; B

_{ij}

_{,s,t}= 1 indicates that the tap position s is lower or equal to the actual position; and B

_{ij}

_{,s,t}= 0 indicates that the tap position s is higher than the actual position. The values of Δτ

_{s}and Δτ

_{s}

^{2}for different tap position s are shown in Table 2.

_{ij}

_{,s,t},u

_{j}

_{,t}, so we introduce λ

_{ij}

_{,s,t}= B

_{ij}

_{,s,t},u

_{j}

_{,t}. Using the big-M relaxation technique, (22) becomes:

## 3. The PWCH Uncertainty Set

#### 3.1. The Convex Hull Uncertainty Set

**u**

_{i}= [u

_{i}

_{,1}, u

_{i}

_{,2}, …, u

_{i}

_{,D}]

^{T}∈ℝ

^{D}, i.e., a point in the D-dimensional Euclidean space. Then, the N historical scenarios can be represented as a high-dimensional point set Ω

_{u}= {u

_{1}, u

_{2}, …, u

_{N}}, consisting of N points in the D-dimensional Euclidean space. A high-dimensional convex hull enclosing all the points can be constructed as:

#### 3.2. The PWCH Uncertainty Set

**u**and

**u**

^{(m,n)}are the scenario vectors in the D-dimensional Euclidean space and the 2-D vectors after projection in the 2D plane corresponding to the mth and the nth dimension, respectively, and

**A**

^{(m,n)}and

**b**

^{(m,n)}are the coefficient matrices and the right-side vector of the to the linear inequality constraint for the corresponding 2D convex hull, respectively.

## 4. Two-Stage Robust Optimal Scheduling for FDNs

#### 4.1. The Two-Stage Framework

#### 4.2. Mathematical Formulation

**x**is the first-stage control variable vector;

**y**is the joint vector of the second-stage control variables and the second-stage state variables;

**d**is the scenario variable vector; L(

**x**,

**d**) is the objective function under the first-stage decision

**x**and the scenario

**d**; $\mathcal{X}$ is the set of all feasible day-ahead decisions, including OLTC action strategies and ESS charging/discharging power during each period; and $\mathcal{Y}(x,d)$ is the set of all feasible second-stage solutions under the first-stage decision

**x**and the scenario

**d**, defined as

**D**,

**f**and

**A**are the coefficient matrices and right-side vector after all linear inequality constraints are rewritten into the matrix-vector form;

**C**is the coefficient matrix after all linear equality constraints are rewritten into the matrix-vector form; and

**G**and

**g**are the coefficient matrix and vector after all second-order cone constraints are rewritten into the matrix-vector form.

#### 4.3. Solution Algorithm

**b**is the coefficient vector after the objective function is rewritten in the matrix-vector form;

**π**,

**λ**, $\sigma $ and

**μ**are the Lagrange multiplier (dual variables) vectors of the corresponding constraints; and the superscript * indicates the results of the master problem.

#### 4.4. Subproblem Solution Algorithm

**d**

^{T}

**π**in the objective function of model (33). For the nonconvex bilinear term, although the Gurobi 9 solver succeeds in solving the model solution through the spatial branching method, it is still rather time-consuming. Therefore, we decompose the subproblem into a linear programming (LP) problem and a second-order cone programming (SOCP) problem by alternating direction iteration of auxiliary and dual variables.

**d*** =

**d**

_{0}. Set the upper bound of the objective function as a larger number.

**π**,

**λ**,

**δ**and

**µ**), solve the following inner LP problem, update the upper bound of the inner loop, and pass the solved auxiliary variable

**d**to the inner SOCP problem:

**d**, solve the following inner SOCP problem, update the lower bound of the inner loop and pass the solved dual variables (

**π**,

**λ**,

**δ**and

**μ**) to the inner LP problem:

## 5. Results

#### 5.1. The 33-Bus Distribution Network

#### 5.1.1. Simulation Settings

#### 5.1.2. Worst-Case Scenario Analysis

#### 5.1.3. OLTC Scheduling Strategy Analysis

#### 5.1.4. Analysis of Bus Voltage Levels

#### 5.1.5. FDS Scheduling Strategy Analysis

#### 5.1.6. Daily Network Loss Comparison

#### 5.2. A Realistic 104-Bus Distribution Network

#### 5.2.1. Simulation Settings

#### 5.2.2. OLTC/ESS Scheduling Strategy Analysis

#### 5.2.3. Analysis of Bus Voltage Levels

#### 5.2.4. FDS Scheduling Strategy Analysis

#### 5.2.5. Daily Network Loss Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**Bus voltage levels using two scheduling methods. (

**a**) Box uncertainty set; (

**b**) PWCH uncertainty set.

**Figure 13.**SOC comparison of the ESSs using two scheduling methods (the 104-bus system). (

**a**) ESS 1 at bus 7; (

**b**) ESS 2 at bus 56; (

**c**) ESS 3 at bus 76; (

**d**) ESS 4 at bus 90.

**Figure 14.**Bus voltage levels using two scheduling methods (the 104-bus system). (

**a**) Box uncertainty set; (

**b**) PWCH uncertainty set.

**Figure 15.**FDS power strategies using two scheduling methods (the 104-bus system). (

**a**) real power; (

**b**) reactive power.

Reference | First Stage Decision | Second Stage Decision | Uncertainty Set | Solution Algorithm |
---|---|---|---|---|

[5] | Network topology | \ | Polyhedron | CCG |

[6] | Network topology | \ | Polyhedron | CCG |

[7] | Network topology, reactive output of VAR compensators and OLTC ratios | DG installation capacity | Polyhedron | CCG |

[8] | OLTC ratios, discrete VAR compensators and charge–discharge power of ESSs | Continuous VAR compensators | Box | CCG |

[9] | Startup/shutdown state of diesel engine generator, operating state of the converters | Individual units | Polyhedron | CCG |

[10] | Slopes of power droop control of VSCs | \ | Box | CCG |

[11] | Power injection of soft open points | \ | Polyhedron | CCG |

[12] | Network topology and power injection of soft open points | Power injection of soft open points | Box | CCG |

[13] | Switching capacitor and OLTC ratios | SVG | Data-adaptive polyhedron | Extreme Scenario |

[15] | Non-AGC units | AGC units | PWCH | CG |

[16] | Capacitor banks and OLTC ratios | PV inverters | Polyhedron | CCG |

s | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

τ_{s} | 0.95 | 0.96 | 0.97 | 0.98 | 0.99 | 1 | 1.01 | 1.02 | 1.03 | 1.04 | 1.05 |

Δτ_{s} | / | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 |

τ_{s}^{2} | 0.9025 | 0.9216 | 0.9409 | 0.9604 | 0.9801 | 1 | 1.201 | 1.404 | 1.609 | 1.0816 | 1.1025 |

Δτ_{s}^{2} | / | 0.0191 | 0.0193 | 0.0195 | 0.0197 | 0.0199 | 0.0201 | 0.0203 | 0.0205 | 0.0207 | 0.0209 |

Time Period | PV1 | PV2 | PV3 | PV4 | PV5 |
---|---|---|---|---|---|

8 | 0.684169 | 0.681293 | 0.679652 | 0.675548 | 0.671436 |

9 | 1.227029 | 1.227765 | 1.213053 | 1.221142 | 1.196871 |

10 | 1.552959 | 1.558479 | 1.534559 | 1.548357 | 1.50604 |

11 | 1.746943 | 1.741806 | 1.731539 | 1.742832 | 1.698664 |

12 | 1.950833 | 1.902491 | 1.940068 | 1.947342 | 1.883357 |

13 | 2.171167 | 2.089821 | 2.13277 | 2.068701 | 2.037691 |

14 | 1.655152 | 1.650435 | 1.668362 | 1.609861 | 1.563627 |

15 | 1.37165 | 1.347866 | 1.36587 | 1.360097 | 1.320059 |

16 | 0.868731 | 0.86399 | 0.869861 | 0.867248 | 0.851022 |

Time Period | PV1 | PV2 | PV3 | PV4 | PV5 |
---|---|---|---|---|---|

8 | 0.708400 | 0.708400 | 0.708400 | 0.708400 | 0.708400 |

9 | 1.268833 | 1.268833 | 1.268833 | 1.268833 | 1.268833 |

10 | 1.587000 | 1.587000 | 1.587000 | 1.587000 | 1.587000 |

11 | 1.771767 | 1.771767 | 1.771767 | 1.771767 | 1.771767 |

12 | 2.008667 | 2.008667 | 2.008667 | 2.008667 | 2.008667 |

13 | 2.246333 | 2.246333 | 2.246333 | 2.246333 | 2.246333 |

14 | 1.627633 | 1.627633 | 1.627633 | 1.627633 | 1.627633 |

15 | 1.344733 | 1.344733 | 1.344733 | 1.344733 | 1.344733 |

16 | 0.838733 | 0.838733 | 0.838733 | 0.838733 | 0.838733 |

PWCH | Box | |
---|---|---|

Objective function (MW) | 2.783 | 3.695 |

CPU time (s) | 483 | 511 |

Iterations number | 2 | 2 |

ESS loss (MW) | 0.175 | 0.195 |

FDS loss (MW) | 0.492 | 0.513 |

Network loss (MW) | 2.115 | 2.987 |

PWCH | Box | |
---|---|---|

Objective function (MW) | 4.0734 | 4.3914 |

Iterations number | 2 | 2 |

ESS loss (MW) | 0.0281 | 0.0823 |

FDS loss (MW) | 0.1713 | 0.2976 |

Network loss (MW) | 3.8740 | 4.0115 |

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## Share and Cite

**MDPI and ACS Style**

Yang, H.; Yuan, S.; Wang, Z.; Liang, D. Two-Stage Robust Optimal Scheduling of Flexible Distribution Networks Based on Pairwise Convex Hull. *Sustainability* **2023**, *15*, 6093.
https://doi.org/10.3390/su15076093

**AMA Style**

Yang H, Yuan S, Wang Z, Liang D. Two-Stage Robust Optimal Scheduling of Flexible Distribution Networks Based on Pairwise Convex Hull. *Sustainability*. 2023; 15(7):6093.
https://doi.org/10.3390/su15076093

**Chicago/Turabian Style**

Yang, Haiyue, Shenghui Yuan, Zhaoqian Wang, and Dong Liang. 2023. "Two-Stage Robust Optimal Scheduling of Flexible Distribution Networks Based on Pairwise Convex Hull" *Sustainability* 15, no. 7: 6093.
https://doi.org/10.3390/su15076093